Volume of revolution about the y-axis (shells)

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The region bounded by $y=x^{2}$, the $x$-axis and $x=2$ is revolved about the $y$-axis. Compute the volume using cylindrical shells.

Solution

Shell of radius $x$, height $x^{2}$, thickness $dx$: $V=2\pi\displaystyle\int_0^2 x^{3}dx=2\pi\Big[\tfrac{x^{4}}{4}\Big]_0^2=8\pi\approx25.133$. $\boxed{V=8\pi\approx25.133}$